Introduction

CHAPTER 1

Introduction

1. Overview

Quantum field theory has been wildly successful as a framework for the study of high-energy particle physics. In addition, the ideas and techniques of quantum field theory have had a profound influence on the development of mathematics.

There is no broad consensus in the mathematics community, however, as to what quantum ﬁeld theory actually is.

This book develops another point of view on perturbative quantum ﬁeld theory, based on a novel axiomatic formulation.

Most axiomatic formulations of quantum field theory in the literature start from the Hamiltonian formulation of field theory. Thus, the Segal [Seg99] axioms for field theory propose that one assigns a Hilbert space of states to a closed Riemannian manifold of dimension d 1, and a unitary operator between Hilbert spaces to a d dimen- − sional manifold with boundary. In the case when the d dimensional manifold is of the form M [0, t], we should view the corresponding operator as time evolution. × The Haag-Kastler [Haa92] axioms also start from the Hamiltonian formulation, but in a slightly different way. They take as the primary object not the Hilbert space, but rather a C! algebra, which will act on a vacuum Hilbert space.

I believe that the Lagrangian formulation of quantum ﬁeld theory, using Feyn- man’s sum over histories, is more fundamental. The axiomatic framework developed in this book is based on the Lagrangian formalism, and on the ideas of low- energy effective ﬁeld theory developed by Kadanoff [Kad66], Wilson [Wil71], Polchin- ski [Pol84] and others.

1.1. The idea of the deﬁnition of quantum ﬁeld theory I use is very simple. Let us assume that we are limited, by the power of our detectors, to studying physical

7 8 1. INTRODUCTION phenomena that occur below a certain energy, say Λ. The part of physics that is visible to a detector of resolution Λ we will call the low-energy effective ﬁeld theory. This low-energy effective ﬁeld theory is succinctly encoded by the energy Λ version of the Lagrangian, which is called the low-energy effective action Se f f [Λ].

The notorious inﬁnities of quantum ﬁeld theory only occur if we consider phenomena of arbitrarily high energy. Thus, if we restrict attention to phenomena occurring at energies less than Λ, we can compute any quantity we would like in terms of the effective action Se f f [Λ].

If Λ# < Λ, then the energy Λ# effective ﬁeld theory can be deduced from knowl- edge of the energy Λ effective ﬁeld theory. This leads to an equation expressing the e f f e f f scale Λ# effective action S [Λ#] in terms of the scale Λ effective action S [Λ]. This equation is called the renormalization group equation.

If we do have a continuum quantum field theory (whatever that is!) we should, in particular, have a low-energy effective field theory for every energy. This leads to our definition : a continuum quantum field theory is a sequence of low-energy effective actions Se f f [Λ], for all Λ < , which are related by the renormalization group flow. In addition, we require that the Se f f [Λ] satisfy a locality axiom, which says that the effective actions Se f f [Λ] become∞ more and more local as Λ . → This definition aims to be as parsimonious as possible. The∞only assumptions I am making about the nature of quantum field theory are the following:

(1) The action principle: physics at every energy scale is described by a La- grangian, according to Feynman’s sum-over-histories philosophy. (2) Locality: in the limit as energy scales go to inﬁnity, interactions between ﬁelds occur at points.

1.2. In this book, I develop complete foundations for perturbative quantum ﬁeld theory in Riemannian signature, on any manifold, using this deﬁnition.

The first significant theorem I prove is an existence result: there are as many quantum field theories, using this definition, as there are Lagrangians.

Let me state this theorem more precisely. Throughout the book, I will treat h¯ as a formal parameter; all quantities will be formal power series in h¯ . Setting h¯ to zero amounts to passing to the classical limit. 1. OVERVIEW 9

Let us fix a classical action functional Scl on some space of fields E , which is assumed to be the space of global sections of a vector bundle on a manifold M1. Let T (n)(E , Scl) be the space of quantizations of the classical theory that are defined modulo h¯ n+1. Then,

1.2.1 Theorem. T (n+1)(E , Scl) T (n)(E , Scl) → is a torsor for the abelian group of Lagrangians under addition (modulo those Lagrangians which are a total derivative).

Thus, any quantization deﬁned to order n in h¯ can be lifted to a quantization de- ﬁned to order n + 1 in h¯ , but there is no canonical lift; any two lifts differ by the addition of a Lagrangian.

If we choose a section of each torsor T (n+)(E , Scl) T (n)(E , Scl) we ﬁnd an → isomorphism T ( )(E , Scl) = series Scl + h¯ S(1) + h¯ 2 S(2) + ∼ · · · where each S(i) is a local∞ functional, that is, a functional which can be written as the integral of a Lagrangian. Thus, this theorem allows one to quantize the theory associated to any classical action functional Scl. However, there is an ambiguity to quantization: at each term in h¯ , we are free to add an arbitrary local functional to our action.

1.3. The main results of this book are all stated in the context of this theorem.

In Chapter 4, I give a deﬁnition of an action of the group R>0 on the space of theories on Rn. This action is called the local renormalization group ﬂow, and is a fundamental part of the concept of renormalizability developed by Wilson and others. The n action of group R>0 on the space of theories on R simply arises from the action of this group on Rn by rescaling.

The coefﬁcients of the action of this local renormalization group ﬂow on any particular theory are the functions of that theory. I include explicit calculations of the function of some simple theories, including the 4 theory on R4.

This local renormalization group ﬂow leads to a concept of renormalizability. Fol- lowing Wilson and others, I say that a theory is perturbatively renormalizable if it has “critical” scaling behaviour under the renormalization group ﬂow. This means that

1The classical action needs to satisfy some non-degeneracy conditions 10 1. INTRODUCTION the theory is fixed under the renormalization group flow except for logarithmic cor- rections. I then classify all possible renormalizable scalar field theories, and find the expected answer. For example, the only renormalizable scalar field theory in four dimensions, invariant under isometries and under the transformation , is the → − 4 theory.

In Chapter 5, I show how to include gauge theories in my definition of quantum field theory, using a natural synthesis of the Wilsonian effective action picture and the Batalin-Vilkovisky formalism. Gauge symmetry, in our set up, is expressed by the requirement that the effective action Se f f [Λ] at each energy Λ satisfies a certain scale Λ Batalin-Vilkovisky quantum master equation. The renormalization group flow is compatible with the Batalin-Vilkovisky quantum master equation: the flow from scale Λ to scale Λ# takes a solution of the scale Λ master equation to a solution to the scale Λ# equation.

I develop a cohomological approach to constructing theories which are renormalizable and which satisfy the quantum master equation. Given any classical gauge theory, satisfying the classical analog of renormalizability, I prove a general theorem allowing one to construct a renormalizable quantization, providing a certain cohomology group vanishes. The dimension of the space of possible renormalizable quantizations is given by a different cohomology group.

In Chapter 6, I apply this general theorem to prove renormalizability of pure Yang- Mills theory. To apply the general theorem to this example, one needs to calculate the cohomology groups controlling obstructions and deformations. This turns out to be a lengthy (if straightforward) exercise in Gel’fand-Fuchs Lie algebra cohomology.

Thus, in the approach to quantum ﬁeld theory presented here, to prove renormalizability of a particular theory, one simply has to calculate the appropriate cohomology groups. No manipulation of Feynman graphs is required.

2. Functional integrals in quantum ﬁeld theory

Let us now turn to giving a detailed overview of the results of this book.

First I will review, at a basic level, some ideas from the functional integral point of view on quantum ﬁeld theory. 2. FUNCTIONAL INTEGRALS IN QUANTUM FIELD THEORY 11

2.1. Let M be a manifold with a metric of Lorentzian signature. We will think of M as space-time. Let us consider a quantum field theory of a single scalar field : M R. → The space of fields of the theory is C (M). We will assume that we have an action functional of the form ∞ S( ) = L ( )(x) x M ! ∈ where L ( ) is a Lagrangian. A typical Lagrangian of interest would be

L ( ) = 1 (D +m2) + 1 4 − 2 4! where D is the Lorentzian analog of the Laplacian operator.

A ﬁeld C (M, R) can describes one possible history of the universe in this ∈ simple model. ∞

Feynman’s sum-over-histories approach to quantum ﬁeld theory says that the universe is in a quantum superposition of all states C (M, R), each weighted by iS( )/h¯ ∈ e . ∞

An observable – a measurement one can make – is a function

O : C (M, R) C. → ∞ If x M, we have an observable O deﬁned by evaluating a ﬁeld at x: ∈ x

Ox( ) = (x).

More generally, we can consider observables that are polynomial functions of the val- ues of and its derivatives at some point x M. Observables of this form can be ∈ thought of as the possible observations that an observer at the point x in the space- time manifold M can make.

The fundamental quantities one wants to compute are the correlation functions of a set of observables, deﬁned by the heuristic formula

iS( )/h¯ O1, . . . , On = e O1( ) On( )D . ' ( C (M) · · · ! ∈ Here D is the (non-existent!) Lebesgue∞ measure on the space C (M).

∞ The non-existence of a Lebesgue measure (i.e. a non-zero translation invariant measure) on an infinite dimensional vector space is one of the fundamental difficulties of quantum field theory. 12 1. INTRODUCTION

We will refer to the picture described here, where one imagines the existence of a Lebesgue measure on the space of ﬁelds, as the naive functional integral picture. Since this measure does not exist, the naive functional integral picture is purely heuristic.

2.2. Throughout this book, I will work in Riemannian signature, instead of the more physical Lorentzian signature. Quantum ﬁeld theory in Riemannian signature can be interpreted as statistical ﬁeld theory, as I will now explain.

Let M be a compact manifold of Riemannian signature. We will take our space of ﬁelds, as before, to be the space C (M, R) of smooth functions on M. Let S : C (M, R) R be an action functional,∞which, as before, we assume is the integral of → 4 a ∞Lagrangian. Again, a typical example would be the action

1 2 1 4 S( ) = 2 (D +m ) + 4! . − x M ! ∈ Here D denotes the non-negative Laplacian. 2

We should think of this field theory as a statistical system of a random field ∈ C (M, R). The energy of a configuration is S( ). The behaviour of the statistical system∞ depends on a temperature parameter T: the system can be in any state with probability S( )/T e− .

The temperature T plays the same role in statistical mechanics as the parameter h¯ plays in quantum ﬁeld theory.

I should emphasize that time evolution does not play a role in this picture: quantum ﬁeld theory on d dimensional space-time is related to statistical ﬁeld theory on d dimensional space. We must assume, however, that the statistical system is in equilib- rium.

As before, the quantities one is interested in are the correlation functions between observables, which one can write (heuristically) as

S( )/T O1, . . . , On = e− O1( ) On( )D . ' ( C (M) · · · ! ∈ ∞ The only difference between this picture and the quantum ﬁeld theory formulation is that we have replaced ih¯ by T.

2Our conventions are such that the quadratic part of the action is negative-deﬁnite. 3. WILSONIAN LOW ENERGY THEORIES 13

If we consider the limiting case, when the temperature T in our statistical system is zero, then the system is “frozen” in some extremum of the action functional S( ). In the dictionary between quantum field theory and statistical mechanics, the zero temperature limit corresponds to classical field theory. In classical field theory, the system is frozen at a solution to the classical equations of motion.

Throughout this book, I will work perturbatively. In the vocabulary of statistical ﬁeld theory, this means that we will take the temperature parameter T to be inﬁnitesi- mally small, and treat everything as a formal power series in T. Since T is very small, the system will be given by a small excitation of an extremum of the action functional.

In the language of quantum ﬁeld theory, working perturbatively means we treat h¯ as a formal parameter. This means we are considering small quantum ﬂuctuations of a given solution to the classical equations of motions.

Throughout the book, I will work in Riemannian signature, but will otherwise use the vocabulary of quantum ﬁeld theory. Our sign conventions are such that h¯ can be identiﬁed with the negative of the temperature.

3. Wilsonian low energy theories

Wilson [Wil71, Wil72], Kadanoff [Kad66], Polchinski [Pol84] and others have stud- ied the part of a quantum ﬁeld theory which is seen by detectors which can only measure phenomena of energy below some ﬁxed Λ. This part of the theory is called the low-energy effective theory.

There are many ways to define “low energy”. I will start by giving a definition which is conceptually very simple, but difficult to work with. In this definition, the low energy fields are those functions on our manifold M which are sums of low-energy eigenvectors of the Laplacian.

In the body of the book, I will use a definition of effective field theory based on length rather than energy. The great advantage of this definition is that it relates better to the concept of locality. I will explain the renormalization group flow from the length-scale point of view shortly.

In this introduction, I will only discuss scalar ﬁeld theories on compact Riemann- ian manifolds. This is purely for expository purposes. In the body of the book I will 14 1. INTRODUCTION work with a general class of theories on a possibly non-compact manifold, although always in Riemannian signature.

3.1. Let M be a compact Riemannian manifold. For any subset I [0, ), let ⊂ C (M) C (M) denote the space of functions which are sums of eigenfunctions I ⊂ of∞the Laplacian∞ with eigenvalue in I. Thus, C (M) Λ denotes the space of functions∞ ≤ that are sums of eigenfunctions with eigenvalue∞ Λ. We can think of C (M) Λ as ≤ ≤ the space of ﬁelds with energy at most Λ. ∞

Detectors that can only see phenomena of energy at most Λ can be represented by functions O : C (M) Λ R[[h¯ ]], ≤ → which are extended to C (M) via the∞ projection C (M) C (M) Λ. → ≤ ∞ ∞ ∞ Let us denote Obs Λ the space of all functions on C (M) that arise in this way. ≤ Elements of Obs Λ will be referred to as observables of ener∞ gy Λ. ≤ ≤ The fundamental quantities of the low-energy effective theory are the correlation functions O1, . . . , On between low-energy observables Oi Obs Λ. It is natural to ' ( ∈ ≤ expect that these correlation functions arise from some kind of statistical system on C (M) Λ. Thus, we will assume that there is a measure on C (M) Λ, of the form ≤ ≤ e f f ∞ eS [Λ]/h¯ D ∞

e f f where D is the Lebesgue measure, and S [Λ] is a function on Obs Λ, such that ≤ Se f f [Λ]( )/h¯ O1, . . . , On = e O1( ) On( )D ' ( C (M) Λ · · · ! ∈ ≤ ∞ for all low-energy observables Oi Obs Λ. ∈ ≤ The function Se f f [Λ] is called the low-energy effective action. This object completely describes all aspects of a quantum ﬁeld theory that can be seen using observables of energy Λ. ≤ Note that our sign conventions are unusual, in that Se f f [Λ] appears in the func- Se f f [Λ]/h¯ Se f f [Λ]/h¯ tional integral via e , instead of e− as is more usual. We will assume the quadratic part of Se f f [Λ] is negative-deﬁnite.

3.2. If Λ Λ, any observable of energy at most Λ is in particular an observable # ≤ # of energy at most Λ. Thus, there are inclusion maps

Obs Λ " Obs Λ ≤ # → ≤ 3. WILSONIAN LOW ENERGY THEORIES 15 if Λ Λ. # ≤

Suppose we have a collection O1, . . . , On Obs Λ of observables of energy at ∈ ≤ # most Λ#. The correlation functions between these observables should be the same whether they are considered to lie in Obs Λ or Obs Λ. That is, ≤ # ≤

e f f S [Λ#]( )/h¯ e O1( ) On( )d C (M) Λ · · · ! ∈ ≤ # ∞ Se f f [Λ]( )/h¯ = e O1( ) On( )d . C (M) Λ · · · ! ∈ ≤ ∞ It follows from this that

e f f 1 e f f S [Λ#]( L) = h¯ log exp S [Λ]( L + H) H C (M) h¯ "! ∈ (Λ#,Λ] # $% ∞ where the low-energy field L is in C (M) Λ . This is a finite dimensional integral, ≤ # and so (under mild conditions) is well∞defined as formal power series in h¯ .

This equation is called the renormalization group equation. It says that if Λ# < Λ, then e f f e f f S [Λ#] is obtained from S [Λ] by averaging over ﬂuctuations of the low-energy ﬁeld L C (M) Λ with energy between Λ# and Λ. ∈ ≤ # ∞

3.3. Recall that in the naive functional-integral point of view, there is supposed to be a measure on the space C (M) of the form

∞ eS( )/h¯ d , where d refers to the (non-existent) Lebesgue measure on the space C (M), and

S( ) is a function of the ﬁeld . ∞

It is natural to ask what role the “original” action S plays in the Wilsonian low- energy picture. The answer is that S is supposed to be the “energy inﬁnity effective action”. The low energy effective action Se f f [Λ] is supposed to be obtained from S by integrating out all ﬁelds of energy greater than Λ, that is

e f f 1 S [Λ]( L) = h¯ log exp S( L + H) . C (M) h¯ "! H ∈ (Λ, ) # $% ∞ This is a functional integral over the inﬁnite dimensional∞ space of ﬁelds with energy greater than Λ. This integral doesn’t make sense; the terms in its Feynman graph expansion are divergent. 16 1. INTRODUCTION

However, one would not expect this expression to be well-defined. The infinite energy effective action should not be defined; one would not expect to have a description of how particles behave at infinite energy. The infinities in the naive functional integral picture arise because the classical action functional S is treated as the infinite energy effective action.

3.4. So far, I have explained how to define a renormalization group equation using the eigenvalues of the Laplacian. This picture is very easy to explain, but it has many disadvantages. The principal disadvantage is that this definition is not local on space-time. Thus, it is difficult to integrate the locality requirements of quantum field theory into this version of the renormalization group flow.

In the body of this book, I will use a version of the renormalization group ﬂow that is based on length rather than on energy. A complete account of this will have to wait until Chapter 2, but I will give a brief description here.

The version of the renormalization group flow based on length is not derived directly from Feynman’s functional integral formulation of quantum field theory. In- stead, it is derived from a different (though ultimately equivalent) formulation of quantum field theory, again due to Feynman [Fey50].

Let us consider the propagator for a free scalar field , with action S f ree( ) = 1 2 Sk( ) = 2 (D +m ) . This propagator P is defined to be the integral kernel for − 2 the inverse of& the operator D +m appearing in the action. Thus, P is a distribution on M2. Away from the diagonal in M2, P is a distribution. The value P(x, y) of P at distinct points x, y in the space-time manifold M can be interpreted as the correlation between the value of the field at x and the value at y.

Feynman realized that the propagator can be written as an integral

m2 P(x, y) = e− K (x, y)d ! ∞=0 where K (x, y) is the heat kernel. The fact that the heat kernel can be interpreted as the transition probability for a random path allows us to write the propagator P(x, y) as an integral over the space of paths in M starting at x and ending at y:

m2 2 P(x, y) = e− f :[0, ] M exp d f . =0 → − 0 + + ! ∞ ! f (0)=x, f ( )=y # ! $ (This expression can be given a rigorous meaning using the Wiener measure). 3. WILSONIAN LOW ENERGY THEORIES 17

From this point of view, the propagator P(x, y) represents the probability that a particle starts at x and transitions to y along a random path (the worldline). The parameter is interpreted as something like the proper time: it is the time measured by a clock travelling along the worldline. (This expression of the propagator is sometimes known as the Schwinger representation).

Any reasonable action functional for a scalar ﬁeld theory can be decomposed into kinetic and interacting terms,

S( ) = S f ree( ) + I( ) where S f ree( ) is the action for the free theory discussed above. From the space-time point of view on quantum ﬁeld theory, the quantity I( ) prescribes how particles interact. The local nature of I( ) simply says that particles only interact when they are at the same point in space-time. From this point of view, Feynman graphs have a very simple interpretation: they are the “world-graphs” traced by a family of particles in space-time moving in a random fashion, and interacting in a way prescribed by I( ).

This point of view on quantum ﬁeld theory is the one most closely related to string theory (see e.g. the introduction to [GSW88]). In string theory, one replaces points by 1-manifolds, and the world-graph of a collection of interacting particles is replaced by the world-sheet describing interacting strings.

3.5. Let us now brieﬂy describe how to treat effective ﬁeld theory from the world- line point of view.

In the energy-scale picture, physics at scales less than Λ is described by saying that we are only allowed ﬁelds of energy less than Λ, and that the action on such ﬁelds is described by the effective action Se f f [Λ].

In the world-line approach, we have, instead of an effective action Se f f [Λ], a scale L effective interaction, Ie f f [L]. This object encodes all physical phenomena occurring at lengths greater than L. One can consider the effective interaction in the energy-scale picture also: the relationship between the effective action Se f f [Λ] and the effective interaction Ie f f [Λ] is simply

e f f 1 e f f S [Λ]( ) = 2 D + I [Λ]( ) − !M 18 1. INTRODUCTION for ﬁelds C (M) . The reason for introducing the effective interaction is that ∈ [0,Λ) the world-line version∞ of the renormalization group ﬂow is better expressed in these terms.

In the world-line picture of physics at lengths greater than L, we can only consider paths which evolve for a proper time greater than L, and then interact via Ie f f [L]. All processes which involve particles moving for a proper time of less than L between interactions are assumed to be subsumed into Ie f f [L].

The renormalization group equation for these effective interactions can be described by saying that quantities we compute using this prescription are independent of L. That is,

3.5.1 Deﬁnition. A collection of effective interactions Ie f f [L] satisﬁes the renormalization group equation if, when we compute correlation functions using Ie f f [L] as our interaction, and allow particles to travel for a proper time of at least L between any two interactions, the result is independent of L.

If one works out what this means, one sees that the scale L effective interaction Ie f f [L] can be constructed in terms of Ie f f [ ] by allowing particles to travel along paths with proper-time between and L, and then interact using Ie f f [ ].

More formally, Ie f f [L] can be expressed as a sum over Feynman graphs, where the edges are labelled by the propagator

L m2 P( , L) = e− K ! and where the vertices are labelled by Ie f f [ ].

This effective interaction Ie f f [L] is a h¯ -dependent functional on the space C (M) e f f of ﬁelds. We can expand I [L] as a formal power series ∞

e f f i e f f I [L] = ∑ h¯ Ii,k [L] i,k 0 ≥ where e f f I [L] : C (M) R i,k → ∞ is homogeneous of order k. Thus, we can think of Ii,k[L] as being a symmetric linear map e f f k I [L] : C (M)⊗ R. i,k → ∞ 3. WILSONIAN LOW ENERGY THEORIES 19

FIGURE 1. The ﬁrst few expressions in the renormalization group ﬂow from scale to scale L. The dotted lines indicate incoming particles. The blobs indicate interactions between these particles. The symbol e f f i Ii,k [L] indicates the h¯ term in the contribution to the interaction of k particles at length-scale L. The solid lines indicate the propagation of a particle between two interactions; particles are allowed to propagate with proper time between and L.

e f f We should think of Ii,k [L] as being a contribution to the interaction of k particles which come together in a region of size around L.

Figure 1 shows how to express, graphically, the world-line version of the renormalization group ﬂow.

3.6. So far in this section, we have sketched the definition of two versions of the renormalization group flow: one based on energy, and one based on length. There is a more general definition of the renormalization group flow which includes these two as special cases. This more general version is based on the concept of parametrix. 20 1. INTRODUCTION

3.6.1 Deﬁnition. A parametrix for the Laplacian D on a manifold is a symmetric distribution P on M2 such that (D 1)P is a smooth function on M2 (where refers to the - ⊗ − M M distribution on the diagonal of M.

This condition implies that the operator Φ : C (M) C (M) associated to the P → kernel P is an inverse for D, up to smoothing operators: both P D Id and D P Id ∞ ∞◦ − ◦ − are smoothing operators.

L If Kt is the heat kernel for the Laplacian D, then Plength(0, L) = 0 Ktdt is a parametrix. This family of parametrices arises when one considers the world-line& picture of quantum ﬁeld theory.

Similarly, we can deﬁne an energy-scale parametrix 1 Penergy[Λ, ) = ∑ e e Λ ⊗ ≥ where the sum is over an orthonormal∞basis of eigenfunctions e for the Laplacian D, with eigenvalue .

Thus, we see that in either the world-line picture or the momentum scale picture one has a family of parametrices ( Plength(0, L) and Penergy[Λ, ), respectively) which converge (in the L 0 and Λ limits, respectively) to the zero distribution. The → → renormalization group equation in either case is written in terms∞ of the one-parameter family of parametrices. ∞

This suggests a more general version of the renormalization group ﬂow, where an arbitrary parametrix P is viewed as deﬁning a “scale” of the theory. In this picture, one e f f should have an effective action I [P] for each parametrix. If P, P# are two different e f f e f f parametrices, then I [P] and I [P#] must be related by a certain renormalization group equation, which expresses Ie f f [P] in terms of a sum over graphs whose vertices are labelled by Ie f f [P ] and whose edges are labelled by P P . # − # If we restrict such a family of effective interactions to parametrices of the form

Plength(0, L) one ﬁnds a solution to the world-line version of the renormalization group equation. If we only consider parametrices of the form Penergy[Λ, ), and then deﬁne

e f f 1 e f f S [Λ]( ) = 2 D + I [Penergy[Λ, )](∞), − !M for C (M) , one ﬁnds a solution to the energy scale version of the renormal- ∈ [0,Λ) ∞ ization group∞ ﬂow. 4. A WILSONIAN DEFINITION OF A QUANTUM FIELD THEORY 21

A general definition of a quantum field theory along these lines is explained in detail Chapter 2, Section 8. This definition is equivalent to one based only on the world-line version of the renormalization group flow, which is the definition used for most of the book.

4. A Wilsonian deﬁnition of a quantum ﬁeld theory

Any detector one could imagine has some finite resolution, and so only probes some low-energy effective theory, described by some Se f f [Λ]. However, one could imagine building detectors of arbitrarily high (but finite) resolution, and so one could imagine probing Se f f [Λ] for arbitrarily high (but finite) Λ.

As is usual in physics, one should only consider those objects which can in principle be observed. Thus, one should say that all aspects of a quantum ﬁeld theory are encoded in its various low-energy effective theories.

Let us make this into a (rough) deﬁnition. A more precise version of this deﬁnition is given later in this introduction; a completely precise version is given in the body of the book.

4.0.2 Deﬁnition. A (continuum) quantum ﬁeld theory is:

(1) An effective action e f f S [Λ] : C (M)[0,Λ] R[[h¯ ]] → for all Λ (0, ). More precisely∞ , Se f f [Λ] should be a formal power series both in ∈ the ﬁeld C (M)[0,Λ] and in the variable h¯ . ∈ e f f (2) Modulo h¯ , each∞∞S [Λ] must be of the form

e f f 1 S [Λ]( ) = 2 D + cubic and higher terms. − !M where D is the positive-deﬁnite Laplacian. (If we want to consider a massive scalar ﬁeld theory, we can replace D by D +m2). e f f e f f (3) If Λ# < Λ, S [Λ#] is determined from S [Λ] by the renormalization group equation (which makes sense in the formal power series setting). (4) The effective actions Se f f [Λ] satisfy a locality axiom, which we will sketch below.

Earlier we sketched several different versions of the renormalization group equation; one based on the world-line formulation of quantum field theory, and one defined by considering arbitrary parametrices for the Laplacian. One gets an equivalent 22 1. INTRODUCTION definition of quantum field theory using either of these versions of the renormalization group flow.

5. Locality

Locality is one of the fundamental principles of quantum ﬁeld theory. Roughly, locality says that any interaction between fundamental particles occurs at a point. Two particles at different points of space-time can not spontaneously affect each other. They can only interact through the medium of other particles. The locality requirement thus excludes any “spooky action at a distance”.

Locality is easily understood in the naive functional integral picture. Here, the theory is supposed to be described by a functional measure of the form eS( )/h¯ d , where d represents the non-existent Lebesgue measure on C (M). In this picture, locality becomes the requirement that the action function S is a∞local action functional. 5.0.3 Deﬁnition. A function S : C (M) R[[h¯ ]] → is a local action functional, if it can be written∞ as a sum

S( ) = ∑ Sk( ) where Sk( ) is of the form

Sk( ) = (D1 )(D2 ) (Dk )dVolM !M · · · where Di are differential operators on M.

Thus, a local action functional S is of the form

S( ) = L ( )(x) x M ! ∈ where the Lagrangian L ( )(x) only depends on Taylor expansion of at x.

5.1. Of course, the naive functional integral picture doesn’t make sense. If we want to give a definition of quantum field theory based on Wilson’s ideas, we need a way to express the idea of locality in terms of the finite energy effective actions Se f f [Λ].

As Λ , the effective action Se f f [Λ] is supposed to encode more and more → “fundamental” interactions. Thus, the ﬁrst tentative deﬁnition is the following. ∞ 5. LOCALITY 23

5.1.1 Deﬁnition (Tentative deﬁnition of asymptotic locality). A collection of low-energy effective actions Se f f [Λ] satisfying the renormalization group equation is asymptotically local if there exists a large Λ asymptotic expansion of the form

Se f f [Λ]( ) f (Λ)Θ ( ) / ∑ i i where the Θ are local action functionals. (The Λ limit of Se f f [Λ] does not exist, in i → general). ∞ This asymptotic locality axiom turns out to be a good idea, but with a fundamental problem. If we suppose that Se f f [Λ] is close to being local for some large Λ, then for e f f all Λ# < Λ, the renormalization group equation implies that S [Λ#] is entirely non- local. In other words, the renormalization group ﬂow is not compatible with the idea of locality.

This problem, however, is an artifact of the particular form of the renormalization group equation we are using. The notion of “energy” is very non-local: high-energy eigenvalues of the Laplacian are spread out all over the manifold. Things work much better if we use the version of the renormalization group ﬂow based on length rather than energy.

The length-based version of the renormalization group ﬂow was sketched earlier. It will be described in detail in Chapter 2, and used throughout the rest of the book.

This length scale version of the renormalization group equation is essentially equivalent to the version based on energy, in the following sense:

Any solution to the length-scale RGE can be translated into a solution to the energy-scale RGE and conversely3.

Under this transformation, large length will correspond (roughly) to low energy, and vice-versa.

The great advantage of working with length scales, however, is that one can make sense of locality. Unlike the energy-scale renormalization group flow, the length-scale renormalization group flow diffuses from local to non-local. We have seen earlier that it is more convenient to describe the length-scale version of an effective field theory by an effective interaction Ie f f [L] rather than by an effective action. If the length-scale L

3The converse requires some growth conditions on the energy-scale effective actions Se f f [Λ]. 24 1. INTRODUCTION effective interaction Ie f f [L] is close to being local, then Ie f f [L + ] is slightly less local, and so on.

As L 0, we approach more “fundamental” interactions. Thus, the locality axiom → should say that Ie f f [L] becomes more and more local as L 0. Thus, one can correct → the tentative deﬁnition asymptotic locality to the following:

5.1.2 Deﬁnition (Asymptotic locality). A collection of low-energy effective actions Ie f f [L] satisfying the length-scale version of the renormalization group equation is asymptotically local if there exists a small L asymptotic expansion of the form

Se f f [L]( ) f (L)Θ ( ) / ∑ i i where the Θ are local action functionals. (The actual L 0 limit will not exist, in general). i → Because solutions to the length scale and energy scale RGEs are in bijection, this deﬁnition applies to solutions to the energy scale RGE as well.

We can now update our deﬁnition of quantum ﬁeld theory:

5.1.3 Deﬁnition. A (continuum) quantum ﬁeld theory is:

(1) An effective action

e f f S [Λ] : C (M)[0,Λ] R[[h¯ ]] → ∞ for all Λ (0, ). More precisely, Se f f [Λ] should be a formal power series both in ∈ the ﬁeld C (M)[0,Λ] and in the variable h¯ . ∈ e f f (2) Modulo h¯ , each∞∞S [Λ] must be of the form

e f f 1 S [Λ]( ) = 2 D + cubic and higher terms. − !M where D is the positive-deﬁnite Laplacian. (If we want to consider a massive scalar ﬁeld theory, we can replace D by D +m2). e f f e f f (3) If Λ# < Λ, S [Λ#] is determined from S [Λ] by the renormalization group equation (which makes sense in the formal power series setting). (4) The effective actions Se f f [Λ], when translated into length-scale terms, satisfy the asymptotic locality axiom.

Since solutions to the energy and length-scale versions of the RGE are equivalent, one can base this deﬁnition entirely on the length-scale version of the RGE. We will do this in the body of the book. 6. THE MAIN THEOREM 25

Earlier we sketched a very general form of the RGE, which uses an arbitrary parametrix to define a “scale” of the theory. In Chapter 2, Section 8, we will give a definition of a quantum field theory based on arbitrary parametrices, and we will show that this definition is equivalent to the one described above.

6. The main theorem

Now we are ready to state the ﬁrst main result of this book.

Theorem A. Let T (n) denote the set of theories deﬁned modulo h¯ n+1. Then, T (n+1) is a principal bundle over T (n) for the abelian group of local action functionals S : C (M) R. → ∞ Recall that a functional S is a local action functional if it is of the form S( ) = L ( ) !M where L is a Lagrangian. The abelian group of local action functionals is the same as that of Lagrangians up to the addition of a Lagrangian which is a total derivative.

Choosing a section of each principal bundle T (n+1) T (n) yields an isomor- → phism between the space of theories and the space of series in h¯ whose coefﬁcients are local action functionals.

A variant theorem allows one to get a bijection between theories and local action functionals, once one has made an additional universal (but unnatural) choice.

6.0.4 Deﬁnition. A renormalization scheme is a way to extract the singular part of certain functions of one variable.

More precisely, there is a subalgebra P((0, )) C ((0, )) of functions f ( ), arising ⊂ from variations of Hodge structure. ∞ ∞ ∞ A renormalization scheme is a subspace P((0, )) P((0, )) <0 ⊂ of “purely singular” functions, complementary to the subspace ∞ ∞ P((0, )) 0 P((0, )) ≥ ⊂ of functions whose r limit exists. → ∞ ∞ The choice of a renormalization∞ scheme gives us a way to extract the singular part of functions in P((0, )).

∞ 26 1. INTRODUCTION

The variant theorem is the following.

Theorem B. The choice of a renormalization scheme leads to a bijection between the space of theories and the space of local action functionals

S : C (M) R[[h¯ ]]. → ∞ Equivalently, there is a bijection between the space of theories and the space of Lagrangians up to the addition of a Lagrangian which is a total derivative.

Theorem B implies theorem A, but theorem A is the more natural formulation.

There are certain caveats:

(1) Like the effective actions Se f f [Λ], the local action functional S is a formal power series both in C (M) and in h¯ . ∈ (2) Modulo h¯ , we require that S∞is of the form

1 S( ) = 2 D + cubic and higher terms. − !M There is a more general formulation of this theorem, where the space of ﬁelds is allowed to be the space of sections of a graded vector bundle. In the more general formulation, the action functional S must have a quadratic term which is elliptic in a certain sense.

6.1. Let me sketch how to prove theorem A. Given the action S, we construct the low-energy effective action Se f f [Λ] by renormalizing a certain functional integral. The formula for the functional integral is

e f f S( L+ H )/h¯ S [Λ]( L) = h¯ log e . C (M) '! H ∈ (Λ, ) ( ∞ This expression is the renormalization group ﬂow fr∞ om inﬁnite energy to energy Λ.

This is an inﬁnite dimensional integral, as the ﬁeld H has unbounded energy.

This functional integral is renormalized using the technique of counterterms. This involves first introducing a regulating parameter r into the functional integral, which tames the singularities arising in the Feynman graph expansion. One choice would be to take the regularized functional integral to be an integral only over the finite dimensional space of fields C (M) . ∈ (Λ,r] ∞ 7. RENORMALIZABILITY 27

Sending r recovers the original integral. This limit won’t exist, but one renor- → malizes this limit by introducing counterterms. Counterterms are functionals SCT(r, ) of both r and the ∞ﬁeld , such that the limit

1 1 CT lim exp S( L + H) S (r, L + H) r C (M) h¯ − h¯ → ! H ∈ (Λ,r] # $ ∞ exists. These counterterms∞ depend on the choice of a renormalization scheme.

The effective action Se f f [Λ] is then deﬁned by this limit:

e f f 1 1 CT S [Λ]( L) = lim exp S( L + H) S (r, L + H) r C (M) h¯ − h¯ → ! H ∈ (Λ,r] # $ ∞ ∞ 6.2. In practise, we don’t use the energy-scale regulator r but rather a length-scale regulator . The reason is the same as before: it is easier to construct local theories using the length-scale regulator than the energy-scale regulator. In what follows, I will ignore this rather technical point; to make the following discussion completely accurate, the reader should replace the energy-scale regulator r by the length-scale regulator we will use later.

The counterterms SCT are constructed by a simple inductive procedure, and are local action functionals of the field C (M). ∈ ∞ Once we have chosen such a renormalization scheme, we find a set of counterterms SCT(r, ) for any local action functional S. These counterterms are uniquely determined by the requirements that firstly, the r limit above exists, and secondly, → they are purely singular as a function of the regulating parameter r. ∞ 6.3. What we see from this is that the bijection between theories and local action functionals is not canonical, but depends on the choice of a renormalization scheme. Thus, theorem A is the most natural formulation: there is no natural bijection between theories and local action functionals. Theorem A implies that the space of theories is an infinite dimensional manifold, modelled on the topological vector space of R[[h¯ ]]- valued local action functionals on C (M).

∞

7. Renormalizability

We have seen that the space of theories is an inﬁnite dimensional manifold, modelled on the space of R[[h¯ ]]-valued local action functionals on C (M). ∞ 28 1. INTRODUCTION

A physicist would find this unsatisfactory. Because the space of theories is infinite dimensional, to specify a particular theory, it would take an infinite number of experiments. Thus, we can’t make any predictions.

We need to ﬁnd a natural ﬁnite-dimensional submanifold of the space of all theories, consisting of “well-behaved” theories. These well-behaved theories will be called renormalizable.

7.1. An old-fashioned viewpoint is the following:

A local action functional (or Lagrangian) is renormalizable if it has only ﬁnitely many counterterms:

CT CT S (r) = ∑ fi(r)Si f inite

In general, this definition picks out a finite dimensional subspace of the infinite dimensional space of theories. However, it is not natural: the specific counterterms will depend on the choice of renormalization scheme, and therefore this definition may depend on the choice of renormalization scheme.

More fundamentally, any deﬁnitions one makes should be directly in terms of the only physical quantities one can measure, namely the low-energy effective actions Se f f [Λ]. Thus, we would like a deﬁnition of renormalizability using only the Se f f [Λ].

The following is the basic idea of the deﬁnition we suggest, following Wilson and others.

7.1.1 Definition (Rough definition). A theory, defined by effective actions Se f f [Λ], is renormalizable if the Se f f [Λ] don’t grow too fast as Λ . However, we must measure Se f f [Λ] → in units appropriate to energy scale Λ. ∞

For instance, if Se f f [1] is measured in joules, then Se f f [103] should be measured in kilo-joules, and so on.

However, this change of units only makes sense on Rn. Since we can identify en- 2 n ergy with length− , changing the units of energy amounts to rescaling R . In addition, the ﬁeld C (Rn) can have its own energy (which should be thought of as giving ∈ n the target of the∞map : R R some weight). Once we incorporate both of these → factors, the procedure of changing units (in a scalar ﬁeld theory) is implemented by 7. RENORMALIZABILITY 29 the map

n n Rl : C (R ) C (R ) → ∞ 1∞n/2 1 (x) l − (l− x). 0→

7.2. As our definition of renormalizability only makes sense on Rn, we will now restrict to considering scalar field theories on Rn. We want to measure Se f f [Λ] as Λ , after we have changed units. Define RG (Se f f [Λ]) by → l e f f e f f 2 ∞ RGl(S [Λ])( ) = S [l− Λ](Rl( ))

e f f e f f 2 Thus, RGl(S [Λ]) is the effective action S [l Λ], but measured in units that have been rescaled by l.

We can use the map RGl to implement precisely the deﬁnition of renormalizability suggested above.

7.2.1 Deﬁnition. A theory Se f f [Λ] is renormalizable if RG (Se f f [Λ]) grows at most loga- { } l rithmically as l 0. →

7.3. It turns out that the map RGl defines a flow on the space of theories. 7.3.1 Lemma. If Se f f [Λ] satisfies the renormalization group equation, then so does RG (Se f f [Λ] . { } { l }

Thus, sending Se f f [Λ] RG (Se f f [Λ]) { } → { l } defines a flow on the space of theories: this is the local renormalization group flow.

Recall that the choice of a renormalization scheme leads to a bijection between the space of theories and Lagrangians. Under this bijection, the local renormalization group ﬂow acts on the space of Lagrangians. The constants appearing in a Lagrangian (the coupling constants) become functions of l; the dependence of the coupling constants on the parameter l is called the function. Renormalizability means these coupling constants have at most logarithmic growth in l.

The local renormalization group ﬂow RG , as l 0, can be interpreted geomet- l → rically as focusing on smaller and smaller regions of space-time, while always using units appropriate to the size of the region one is considering. In energy terms, applying RG as l 0 amounts to focusing on phenomena of higher and higher energy. l → 30 1. INTRODUCTION

The logarithmic growth condition thus says the theory doesn’t break down completely when we probe high-energy phenomena. If the effective actions displayed polynomial growth, for instance, then one would ﬁnd that the perturbative description of the theory wouldn’t make sense at high energy, because the terms in the perturbative expansion would increase with the energy.

7.4. The deﬁnition of renormalizability given above can be viewed as a perturbative approximation to an ideal non-perturbative deﬁnition.

7.4.1 Definition (Ideal definition). A non-perturbative theory is renormalizable if, as we flow the theory under RG and let l 0, we converge to a fixed point. l →

This ﬁxed point, if it exists, would be a scaling limit of the theory; it would neces- sarily be a scale-invariant theory. For instance, it is expected that Yang-Mills theory is renormalizable in this sense, and that the scaling limit is a free theory.

This ideal definition is difficult to make sense of perturbatively (when we treat h¯ as a formal parameter). For instance, suppose a coupling constant c changes to c lh¯ c = c + h¯ c log l + . . . 0→ Non-perturbatively, we might think that h¯ > 0, so that this flow converges to a fixed point. Perturbatively, however, h¯ is a formal parameter, so it appears to have logarithmic growth.

Our perturbative definition can be interpreted as saying that a perturbative theory is renormalizable if, at first sight, it looks like it might be non-perturbatively renormalizable in this sense. For instance, if it contains coupling constants which are of polynomial growth in l, these will probably persist at the non-perturbative level, im- plying that the theory does not converge to a fixed point.

One can make a more refined perturbative definition by requiring that the logarithmic growth which does appear is of the correct sign (thus distinguishing between c lh¯ c and c l h¯ c). This more refined definition leads to asymptotic freedom, which 0→ 0→ − is the statement that a theory converges to a free theory as l 0. →

8. Renormalizable scalar ﬁeld theories

Now that we have a deﬁnition of renormalizability, the next question to ask is: which theories are renormalizable? 8. RENORMALIZABLE SCALAR FIELD THEORIES 31

It turns out to be straightforward to classify all renormalizable scalar ﬁeld theories.

8.1. Suppose we have a local action functional S of a scalar ﬁeld on Rn, and suppose that S is translation invariant. We say that S is of dimension k if

k S(Rl( )) = l S( ).

1 n/2 1 Recall that Rl( )(x) = l − (− lx).

Every translation invariant local action functional S is a ﬁnite sum of terms of dimension. For instance:

D is of dimension 0 4 !R 4 is of dimension 0 4 !R ∂ 3 is of dimension 1 4 !R ∂xi − 2 is of dimension 2 4 !R

Now let us state how one classiﬁes scalar ﬁeld theories, in general.

8.1.1 Theorem. Let R(k)(Rn) denote the space of renormalizable scalar ﬁeld theories on Rn, invariant under translation, deﬁned modulo h¯ n+1.

Then, R(k+1)(Rn) R(k)(Rn) → is a torsor for the vector space of local action functionals S( ) which are a sum of terms of non-negative dimension.

Further, R(0)(Rn) is canonically isomorphic to the space of local action functionals of the form

S( ) = 1 D + cubic and higher terms, of non-negative dimension. 2 n − !R

As before, the choice of a renormalization scheme leads to a section of each of the torsors R(k+1)(Rn) R(k)(Rn), and so to a bijection between the space of renormal- → izable scalar ﬁeld theories and the space of series

1 i 2 D + ∑ h¯ Si − ! 32 1. INTRODUCTION

where each Si is a translation invariant local action functional of non-negative dimension, and S0 is at least cubic.

Applying this to R4, we ﬁnd the following. 8.1.2 Corollary. Renormalizable scalar ﬁeld theories on R4, invariant under SO(4) ! R4 and under , are in bijection with Lagrangians of the form → − L ( ) = a D + b 4 + c 2 for a, b, c R[[h¯ ]], where a = 1 modulo h¯ and b = 0 modulo h¯ . ∈ − 2

More generally, there is a finite dimensional space of non-free renormalizable theories in dimensions n = 3, 4, 5, 6, an infinite dimensional space in dimensions n = 1, 2, and none in dimensions n > 6. ( “Finite dimensional” means as a formal scheme over Spec R[[h¯ ]]: there are only finitely many R[[h¯ ]]-valued parameters).

Thus we find that the scalar field theories traditionally considered to be “renormalizable” are precisely the ones selected by the Wilsonian definition advocated here. However, in this approach, one has a conceptual reason for why these particular scalar field theories, and no others, are renormalizable.

9. Gauge theories

We would like to apply the Wilsonian philosophy to understand gauge theories. In Chapter 5, we will explain how to do this using a synthesis of Wilsonian ideas and the Batalin-Vilkovisky formalism.

9.1. In mathematical parlance, a gauge theory is a field theory where the space of fields is a stack. A typical example is Yang-Mills theory, where the space of fields is the space of connections on some principal G-bundle on space-time, modulo gauge equivalence.

It is important to emphasize the difference between gauge theories and ﬁeld theories equipped with some symmetry group. In a gauge theory, the gauge group is not a group of symmetries of the theory. The theory does not make any sense before taking the quotient by the gauge group.

One can see this even at the classical level. In classical U(1) Yang-Mills theory on a 4-manifold M, the space of ﬁelds (before quotienting by the gauge group) is Ω1(M). 9. GAUGE THEORIES 33

The action is S( ) = d d . The highly degenerate nature of this action means M ∗ that the classical theory& is not predictive: a solution to the equations of motion is not determined by its behaviour on a space-like hypersurface. Thus, classical Yang-Mills theory is not a sensible theory before taking the quotient by the gauge group.

9.2. Let us now discuss gauge theories in effective ﬁeld theory. Naively, one could imagine that to give a gauge theory would be to give an effective gauge theory at every energy level, in a way related by the renormalization group ﬂow.

One immediate problem with this idea is that the space of low-energy gauge symmetries is not a group. The product of low-energy gauge symmetries is no longer low- energy; and if we project this product onto its low-energy part, the resulting multipli- cation on the set of low-energy gauge symmetries is not associative.

For example, if g is a Lie algebra, then the Lie algebra of inﬁnitesimal gauge symmetries on a manifold M is C (M) g. The space of low-energy inﬁnitesimal ⊗ gauge symmetries is then C (M) ∞Λ g. In general, the product of two functions in ≤ ⊗ C (M) Λ can have arbitrary∞energy; so that C (M)<Λ g is not closed under the Lie ≤ ⊗ bracket.∞ ∞

This problem is solved by a very natural union of the Batalin-Vilkovisky formalism and the effective action philosophy.

9.3. The Batalin-Vilkovisky formalism is widely regarded as being the most pow- erful and general way to quantize gauge theories. The first step in the BV procedure is to introduce extra fields – ghosts, corresponding to infinitesimal gauge symmetries; anti-fields dual to fields; and anti-ghosts dual to ghosts – and then write down an extended classical action functional on this extended space of fields.

This extended space of fields has a very natural interpretation in homological algebra: it describes the derived moduli space of solutions to the Euler-Lagrange equations of the theory. The derived moduli space is obtained by first taking a derived quotient of the space of fields by the gauge group, and then imposing the Euler-Lagrange equations of the theory in a derived way. The extended classical action functional on the extended space of fields arises from the differential on this derived moduli space.

In more pedestrian terms, the extended classical action functional encodes the following data: 34 1. INTRODUCTION

(1) the original action functional on the original space of fields; (2) the Lie bracket on the space of infinitesimal gauge symmetries, (3) the way this Lie algebra acts on the original space of fields.

In order to construct a quantum theory, one asks that the extended action satisﬁes the extended action satisﬁes the quantum master equation. This is a succinct way of encoding the following conditions:

(1) The Lie bracket on the space of infinitesimal gauge symmetries satisfies the Jacobi identity. (2) This Lie algebra acts in a way preserving the action functional on the space of fields. (3) The Lie algebra of infinitesimal gauge symmetries also preserves the “Lebesgue measure” on the original space of fields. That is, the vector field on the original space of fields associated to every infinitesimal gauge symmetry is divergence free. (4) The adjoint action of the Lie algebra on itself preserves the “Lebesgue measure”. Again, this says that a vector field associated to every infinitesimal gauge symmetry is divergence free.

Unfortunately, the quantum master equation is an ill-defined expression. The 3rd and 4th conditions above are the source of the problem: the divergence of a vector field on the space of fields is a singular expression, involving the same kind of singularities as those appearing in one-loop Feynman diagrams.

9.4. This form of the quantum master equation violates our philosophy: we should always express things in terms of the effective actions. The quantum master equation above is about the original “inﬁnite energy” action, so we should not be surprised that it doesn’t make sense.

The solution to this problem is to combine the BV formalism with the effective action philosophy. To give an effective action in the BV formalism is to give a functional Se f f [Λ] on the energy Λ part of the extended space of fields (i.e., the space of ghosts, ≤ fields, anti-fields and anti-ghosts). This energy Λ effective action must satisfy a certain energy Λ quantum master equation.

The reason that the effective action philosophy and the BV formalism work well together is the following. 9. GAUGE THEORIES 35

Lemma. The renormalization group ﬂow from scale Λ to scale Λ# carries solutions of the energy Λ quantum master equation into solutions of the energy Λ# quantum master equation.

Thus, to give a gauge theory in the effective BV formalism is to give a collection of effective actions Se f f [Λ] for each Λ, such that Se f f [Λ] satisﬁes the scale Λ QME, e f f e f f and such that S [Λ#] is obtained from S [Λ] by the renormalization group ﬂow. In addition, one requires that the effective actions Se f f [Λ] satisfy a locality axiom, as before.

This picture also solves the problem that the low energy gauge symmetries are not a group. The energy Λ effective action Se f f [Λ], satisfying the energy Λ quantum master equation, gives the extended space of low-energy ﬁelds a certain homotopical algebraic structure, which has the following interpretation:

(1) The space of low-energy infinitesimal gauge symmetries has a Lie bracket. (2) This Lie algebra acts on the space of low-energy fields. (3) The space of low-energy fields has a functional, invariant under the bracket. (4) The action of the Lie algebra on the space of fields, and on itself, preserves the Lebesgue measure.

But, these axioms don’t hold on the nose, but hold up to a sequence of coherent higher homotopies.

9.5. Let us now formalize our definition of a gauge theory. As we have seen, whenever we have the data of a classical gauge theory, we get an extended space of fields, that we will denote by E . This is always the space of sections of a graded vector bundle on the manifold M. As before, let E Λ denote the space of low-energy ≤ extended fields.

9.5.1 Deﬁnition. A theory in the BV formalism consists of a set of low-energy effective actions

e f f S [Λ] : E Λ R[[h¯ ]], ≤ → which is a formal series both in E Λ and h¯ , and which is such that: ≤ (1) The renormalization group equation is satisﬁed. (2) Each Se f f [Λ] satisﬁes the energy Λ quantum master equation. (3) The same locality axiom as before holds. (4) There is one more technical restriction : modulo h¯ , each Se f f [Λ] is of the form Se f f [Λ](e) = e, Qe + cubic and higher terms ' ( 36 1. INTRODUCTION

where , is a certain canonical pairing on E , and Q : E E satisﬁes certain '− −( → ellipticity conditions.

As before, the locality axiom needs to be expressed in length-scale terms.

The main theorem holds in this context also, but in a slightly modified form. If we remove the requirement that the effective actions satisfy the quantum master equation, we find a bijection between theories and local action functionals, depending on the choice of a renormalization scheme, as before. Requiring that the effective actions satisfy the QME leads to a constraint on the corresponding local action functional, which is called the renormalized quantum master equation. This renormalized QME replaces the ill-defined QME appearing in the naive BV formalism.

Physicists often say that a theory satisfying the quantum master equation is free of “gauge anomalies”. In general, an anomaly is a symmetry of the classical theory which fails to be a symmetry of the quantum theory. In my opinion, this terminology is misleading: the gauge group action on the space of ﬁelds is not a symmetry of the theory, but rather an inextricable part of the theory. The presence of gauge anomalies means that the theory does not exist in a meaningful way.

9.6. Renormalizing gauge theories. It is straightforward to generalize the Wilso- nian definition of renormalizability (Definition 7.2.1) to apply to gauge theories in the BV formalism. As before, this definition only works on Rn, because one needs to rescale space-time. This rescaling of space-time leads to a flow on the space of theories, which we call the local renormalization group flow. (This flow respects the quantum master equation). A theory is defined to be renormalizable if it exhibits at most logarithmic growth under the local renormalization group flow.

Now we are ready to state one of the main results of this book.

Theorem. Pure Yang-Mills theory on R4, with coefﬁcients in a simple Lie algebra g, is perturbatively renormalizable.

That is, there exists a theory Se f f [Λ] , which is renormalizable, which satisﬁes the quan- { YM } tum master equation, and which modulo h¯ is given by the classical Yang-Mills action.

The moduli space of such theories is isomorphic to h¯ R[[h¯ ]]. 10. OBSERVABLES AND CORRELATION FUNCTIONS 37

Let me state more precisely what I mean by this. At the classical level (modulo h¯ ) there are no difficulties with renormalization, and it is straightforward to define pure Yang-Mills theory in the BV formalism4. Because the classical Yang-Mills action is con- formally invariant in four dimensions, it is a fixed point of the local renormalization group flow.

One is then interested in quantizing this classical theory in a renormalizable way.

The theorem states that one can do this, and that the set of all such renormalizable quantizations is isomorphic (non-canonically) to h¯ R[[h¯ ]].

This theorem is proved by obstruction theory. A lengthy (but straightforward) calculation in Lie algebra cohomology shows that the group of obstructions to ﬁnding a renormalizable quantization of Yang-Mills theory vanishes; and that the corresponding deformation group is one-dimensional. Standard obstruction theory arguments then imply that the moduli space of quantizations is R[[h¯ ]], as desired.

This calculation uses the following strange “coincidence” in Lie algebra cohomology: although H5(su(3)) is one dimensional, the outer automorphism group of su(3) acts on this space in a non-trivial way. A more direct construction of Yang-Mills theory, not relying on obstruction theory, is desirable.

10. Observables and correlation functions

The key quantities one wants to compute in a quantum ﬁeld theory are the correlation functions between observables. These are the quantities that can be more-or-less directly related to experiment.

The theory of observables and correlation functions is addressed in the work in progress [CG10], written jointly with Owen Gwilliam. In this sequel, we will show how the observables of a quantum ﬁeld theory (in the sense of this book) form a rich algebraic structure called a factorization algebra. The concept of factorization algebra was introduced by Beilinson and Drinfeld [BD04], as a geometric formulation of the axioms of a vertex algebra. The factorization algebra associated to a quantum ﬁeld theory is a complete encoding of the theory: from this algebraic object one can recon- struct the correlation functions, the operator product expansion, and so on.

4For technical reasons, we use a ﬁrst-order formulation of Yang-Mills, which is equivalent to the usual formulation. 38 1. INTRODUCTION

Thus, a proper treatment of correlation functions requires a great deal of prelim- inary work on the theory of factorization algebras and on the factorization algebra associated to a quantum ﬁeld theory. This is beyond the scope of the present work.

11. Other approaches to perturbative quantum ﬁeld theory

Let me finish by comparing briefly the approach to perturbative quantum field theory developed here with others developed in the literature.

11.1. In the last ten years, the perturbative version of algebraic quantum ﬁeld theory has been developed by Brunetti, Dutsch,¨ Fredenhagen, Hollands, Wald and others: see [BF00, BF09, DF01, HW10]. In this work, the authors investigate the problem of constructing a solution to the axioms of algebraic quantum ﬁeld theory in perturbation theory. These authors prove results which have a very similar form to those proved in this book: term by term in h¯ , there is an ambiguity in quantization, described by a certain class of Lagrangians.

The proof of these results relies on a version of the Epstein-Glaser [EG73] construction of counter-terms. This construction of counter-terms relies, like the approach used in this book, on working directly on real space, as opposed to on momentum space. In the Epstein-Glaser approach to renormalization, as in the approach described here, the proof that the counter-terms are local is easy. In contrast, in momentum-space approaches to constructing counter-terms – such as that developed by Bogliubov- Parasiuk[BP57] and Hepp [Hep66] – the problem of constructing local counter-terms involves complicated graph combinatorics.

11.2. Another approach to perturbative quantum ﬁeld theory on Riemannian space time was developed by Hollands [Hol09] and Hollands-Olbermann [HO09]. In this approach the ﬁeld theory is encoded in a vertex algebra on the space-time manifold.

This seems to be philosophically very closely related to my joint work with Owen Gwilliam [CG10], which uses the renormalization techniques developed in this paper to produce a factorization algebra on the space-time manifold. Thus, the quantization results proved by Hollands-Olbermann should be close analogues of the results presented here. ACKNOWLEDGEMENTS 39

11.3. In a lecture at the conference “Renormalization: algebraic, geometric and probabilistic aspects” in Lyon in 2010, Maxim Kontsevich presented an approach to perturbative renormalization which he developed some years before. The output of Kontsevich’s construction is (as in [HO09] and [CG10]) a vertex algebra on the space- time manifold. The form of Kontsevich’s theorem is very similar to the main theorem of this book: order by order in h¯ , the space of possible quantizations is a torsor for an Abelian group constructed from certain Lagrangians. Kontsevich’s work relies on a new construction of counter-terms which, like the Epstein-Glaser construction and the construction developed here, relies on working in real space rather than on momentum space.

Again, it is natural to speculate that there is a close relationship between Kontse- vich’s work and the construction of factorization algebras presented in [CG10].

11.4. Let me ﬁnally mention an approach to perturbative renormalization developed initially by Connes and Kreimer [CK98, CK99], and further developed by (among others) Connes-Marcolli [CM04, CM08b] and van Suijlekom [vS07]. The ﬁrst result of this approach is that the Bogliubov-Parasiuk-Hepp-Zimmerman [BP57, Hep66] algorithm has a beautiful interpretation in terms of the Birkhoff decomposition for loops in a certain pro-algebraic group constructed combinatorially from graphs.

In this book, however, counter-terms have no intrinsic importance: they are simply a technical tool used to prove the main results. Thus, it is not clear to me if there is any relationship between Connes-Kreimer Hopf algebra and the results of this book.

Acknowledgements

Many people have contributed to the material in this book. Without the constant encouragement and insightful editorial suggestions of Lauren Weinberg this project would probably never have been ﬁnished. Particular thanks are also due to Owen Gwilliam, for many discussions which have greatly clariﬁed the ideas in this book; and to Dennis Sullivan, for his encouragement and many helpful comments. I would also like to thank Alberto Cattaneo, Jacques Distler, Mike Douglas, Giovanni Felder, Arthur Greenspoon, Si Li, Pavel Mnev, Josh Shadlen, Yuan Shen, Stefan Stolz, Peter Teichner, and A.J. Tolland. Of course, any errors are solely the responsibility of the author.